Percolation systems away from the critical point

This article reviews some effects of disorder in percolation systems away from the critical density pc. For densities below pc, the statistics of large clusters defines the animals problem. Its relation to the directed animals problem and the Lee–Yang edge singularity problem is described. Rare compact clusters give rise to Griffiths singularities in the free energy of diluted ferromagnets, and lead to a very slow relaxation of magnetization. In biased diffusion on percolation clusters, trapping in dead-end branches leads to asymptotic drift velocity becoming zero for strong bias, and very slow relaxation of velocity near the critical bias field.